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Regularization and distributional derivatives of (x 1 2 +x 2 2 ++X p 2 ) -n in p . (English) Zbl 0578.46034

Summary: Our main aim is to present the value of the distributional derivative

¯ N x 1 k 1 x 2 k 2 ···x p k p ( 1r n ),

where r=(x 1 2 +x 2 2 +···+x p 2 ) 1/2 in p , N=k 1 +k 2 +···+k p , and p,n,k 1 ,k 2 ,···,k p are positive integers. For this purpose, we first define a regularization of 1/x n in 1 , which in turn helps us to define the regularization of 1/r n in p . These regularizations are achieved as asymptotic limits of the truncated functions H(x-ϵ)/x n and H(r-ϵ)/r n as ϵ 0, plus certain terms concentrated at the origin, where H is the Heaviside function. In the process of the derivation of the distributional derivative formula mentioned, we also derive many other interesting results and introduce some simplifying notation.


MSC:
46F10Operations with distributions (generalized functions)